Internat.
J. Math.&
Math. Sci. Vol. i0No.
2(1987)
381-390
RAYLEIGH WAVE SCATTERING
AT
THE
FOOT OF A MOUNTAIN
381
P.S. DESHWAL
andK.K. MANN
Department of Mathematics Maharshi Dayanand University
Rohtak-124001, INDIA (Received March 3, 1986)
ABSTRACT. A theoretical study of scattering of seismic waves at the foot of a mountain
is discussed here. A mountain of an arbitrary shape and of width a (0
x
a, z0)
in the surface of an elastic solid medium (z0)
is hit by a Rayleigh wave The method of solution is the technique of Wiener and Hopf. The reflected, transmitted and scattered waves are obtained by inversion of Fourier transforms. The scattered waves behave as decaying cylindrical waves at distant points and have a large amplitude near the foot of the mountain. The transmitted wave decreases exponentially as its distance from the other end of the mountain increases.KEY WORDS AND PHRASES.
Rayleigh
surface
waves,
wave
motionsin
elasticsolids,
reflected
and transmitted waves.
IgBO
AMS
SUBJECT CLASSIFICATION CODES.
73D20,
?3DIO.I. INTRODUCTION.
Earthquakes throw a grave challenge to human beings. Seismic waves cause the havoc because of their scattering due to the inhomogeneities and discontinuities in the sur-face of the earth. Rayleigh waves, appearing on the surface of the earth, are responsi-ble for destruction of the buildings and loss of human lives. Scattering of seismic waves due to irregularities in the surface leads to large amplification and variation in ground motion during earthquakes.
The paper presents a mechanism by which the energy of a body wave is lessened by partial conversion into reflected body and surface waves and by scattering at the foot of a mountain.
382
medium is a liquid halfspace which reduces the number of elastic parameters, the basic equations and the boundary conditions. Not much theoretical study is available on the problem of scattering of elastic waves due to an irregular boundary of finite dlmenslon Recently Momol (1980, 82) has studied the problem of scattering of Rayleigh waves by semicircular and rectangular discontinuities in the surface of a solid halfspace. The solutions are not exact because of the approximations used in the solution of these problems.
We propose to discuss here the problem of scattering of Raylelgh waves at the foot of a mountain with its base occupying the region 0 x
!
a, z 0 in the surface of a solid halfspace z 0 Since we are interested in scattering of waves at the foot of a mountain, its shape is immaterial and it is assumed to be rigid such that there is no displacement across the mountain. The method of solution is the Fourier transformation of the basic equations and determination of unknown functions by the technique of Wienerand Hopf (Noble,
(1958)).
2. STATEMENT OF THE PROBLEM.The problem is two-dimenslonal in zx-plane. The mountain of width a has its
base along 0 x a, z 0 in the surface of an elastic solid halfspace z 0 (figure
I).
The medium is homogeneous, isotropic and slightly dissipative. If the re-tarding force of the medium is proportional to the velocity, then the wave equation is2
2
2
+
E+
(2.1)
x
2 3z2 c2t
2 c 2t
where c is the velocity of propagation and e > 0 is the damping constant. The pote tial function harmonic in time is
(x,z,t)
(x,z)
exp(-imt)(2.2)
The equation
(2.1)
is now(V
2+
()2)
(x,z)
0(2.3)
where
/ (m2
+
le)/c
I
+
i2
is complex whose imaginary part is small andposi-tive.
An
incident waveiP0X
-B0z
#i(x,z)
D(2p
k2)
e e(2.4)
iP0x
-60z
i(x’z)
D(21P080)
e e(2.5)
where
P0
is a root of the Raylelgh frequency equationf(p)
(2p
2 k2)
4p286
0,8
/
(p2
()2),
6/
(p2
k2)
(2.6)
80
8(p0),
60
6(p0),
strikes at the foot of the mountain from the region x < 0. The potential(x,z)
satisfies the wave equation(V
2+
k2)
(x,z)
0(2.7)
Let the total potentials be
#t(x’z)
(x,z)
+
#i(x’z)
(2.8)
RAYLEIGH
WAVE
SCATTERINGAT THE
FOOT OFA
MOUNTAIN 383 The potentials are connected with the displacements (u,w) byu
t/x
+
t/z
wt/z
t/z
(2.10)
We assume that, for given z, (x,z) and (x,z) have the behaviour
exp(-dlxl)
asIx]
,
d > 0 The Fourier transform_(p,z)
f0_
(x,z)
eipx dx, p a+
ia0
(2.11)
is analytic in the region n
0 < d of the complex p-plane. The Fourier transform ipx
#(p,z)
I
(x,z)
e dx ipxla
ipxI
0 (x,z) e dx+
(x,z)
e dx+
I
(x,z)
eipx dx0 a
#_(p,z) +
#a(P’z)
+
+a(P’Z)
(2.12)
and its derivatives w.r.t.z are analytic in the strip -d < s
0 < d of the complex plane. The transform of
(x,z)
has the same behaviour.3. BOUNDARY CONDITIONS.
The boundary conditions of the problem are
(i) u
#t/x
+
t/z
0 z 0 0 x a,(3.1)
(ii) w
t/Sz
t/x
0 z 0 0!
x!
a,(3.2)
(iii)
Zxkz
x
k2@
0 z 0 x!
0 xZ
a,(3.3)
(iv)
2x,x
+
z)
+
k2
0 z 0 x!
0 x a,(3.4)
The conditions
(3.1)
and(3.2)
subject to (2.3) and(2.7)
reduce toiPoX
-B0z
-i
-D(2p
k2)
e e z 0, 0 x a,(3.5)
iP0x
-60z
-i
-D(ZiP0B0)
e e z 0, 0!
x!
a,(3.6)
4. SOLUTION OF THE PROBLEM.Fourier transforms of the wave equations
(2.3)
and(2.7)
lead tod2Idz
282
0 8+/-/
(p2
(i2)
(4.1)
d2/dz
262
0 6+/-/
(p2
k2(4.2)
Since
(x,z)
and@(x,z)
are bounded as z tends to infinity and their transforms are also bounded. The solutions of(4.1)
and(4.2)
are(p,z)
A(p)
e-8z
(4o3)
@(p,z) B(p) e (4.4)
The signs in radicals for
8
and 6 are such that their real parts are positive for all p. We use the notations(p)
@(p) for (p,0)(p,O)
etc. From (4.3) we obtain384 P.S.
DESHWAL
and K.K.MANN
This equation is decomposed(p)
#(-k)
#’_(-k)
]+
+(P)
+
’
(p)
’
(-k)
+
(k)_
(p)-
(4.6)
p+/k p-/k
/-2
/2(p-)
The left hand member of
(4.6)
is analytic in a0 > -d and the right hand member in
a
0 > d They represent an entire function. Further each member tends to zero as
Pl
By Liouville’s theorem, the entire function is identically zero. Equating each member to zero, it is found that+(p)
-
+(p)/8
+
qF(p),
q(k)
’_(-k)
(4.7)#_(p)
-#’_(p)/8
qF(p),
-q’_(k)
i(-k)
F(p)
/2(p-)
/-2(p+)
Similarly, we find from
(4.4)
that+(p)
-$+(p)/6
+
qG(p)
q$(k)
’_(-k)
(4.8)
(4.9)
(4.10)
@_(p)
=-’(p)16
qG(p) -q@’_(k)
(-k)
(4.11)
G(p)
(4.12)
/2k(p-k)
J-2k(p+k)
Fourier transforms of
(3.5)
and(3.6)
lead toD(2p
k2)
i(p+p0)a
dxi(p
+
p0
(eI)
(4.13)
i(p+P0)X
a
(p)
-m(2p
k2)
fa
e 0a
(p)
-D(2iP0B0)
la
i
(P+P0)
x-2P080D
i(p+P0)
ae dx
(e
i)(4.14)
0
P+
P0
And those of
(3.1)
and(3.2)
result ini
(p+p0)
aa(p)
2mP0B060(e
l)/(p+p0)
(4.15)
i
(p+p0)
a(p)
-imB0(2P
k2)(e
l)/(p+p0)
(4.16)We now multiply
(3.3)
by exp(ipx) and integrate between x and x 0 tofind out
2
f0
3__(8_
ipxk2
f_0
eipX
8x 3z
x
)e
dx dx 0(4.17)
Since w
3#/3z
3/3x
is continuous, the first integral in(4.17)
is integrated by parts to obtain3@)
ipxk2
8
3
-2ip
/_0
(38z
x
e dx_(p)
-2(z
x)0
or8i
3i)
-2ipI(P)
+
(2p2- k2)
-(P)
(3z
x
RAYLEIGH WAVE SCATTERING AT THE
FOOT OFA
MOUNTAIN 385 In the same way,(3.4)
leads to(2p2-k2)_(p)
+
2ip’_(p)
2ip[(p-po)(2p-k2)
+
2P08060
Solving
(4.8)
(4.111
(4.18) and(4.19)
for_(p)
and@_(p),
we find(4.19)
f(p)_(p)
2ip6[q(2p 2- k2)
G(p)
2ipSF(p)]
+
2iD(2p2-k2)[(p-p0)(2p-k2)
+
2P0B00
+4iPBoD(k2-2pp
O)
(4.20)
f(p)_(p)
(2p2-k2)[-2ipBF(p)
+
2BoD(k2-2pp
01]
-2ipB[2ip6qG(p)
+
2iD[(p-p0)(2p-k2)
+
2P08060
]]
(4.21)
Integrating(3.3)
and(3.4)
between x a and x after multiplying it byexp(ipx), it is obtained that
i(p+po)a
(2P
2 -k2)
+a(p)
2ipa(p)
=-2oD(k2-
2pp0)
e(4.22)
(2p2
-k21
La
(p)
+
2ip-a
(p)
-2iD[(p-p0)
(2p-k
21
+
2P0060].
exp (i
(p+p0)
a)(4.23)
Inversion of Fourier transforms will give various waves.5.
VARIOUS
WAVES.The factor
exp(-ipx)
exp(-iex)exp(eoX
in the inverse transforms makes the inte-grals vanish at infinity in the upper part of the complex plane if x < 0 and in the lower part if x > 0.For
waves in the region x < O,we have
$(x
z)
+ieo
-ipx+ie0
$_(p,z)
e dp (5.1)where the line integral
(5.1)
is in the strip -d < < d The contour is in the oupper half of the complex plane where
_(-p,z)
is analytic and hence of
_(-p,z)
e-ipx dp(5.2)
0
o From
(5.1)
and (5.2), it is found that=’+iaO
,(x,z)
+ia0
[0_(p,z)
0_(-p,z)]
eipXdp
(5.3)
To find the integrand, the wave equation(2.3)
is integrated from x to x 0 after multiplying it by exp(ipx), to find outd2_/dz
282_
ffi-($/x)0
+
ip()0(5.4)
Changing p to -p and subtracting the resulting equation from (5.41, we obtain
(d2/dz
2821[_(p,z)
_(-p,z)]
-2ipD(2p-k
21 exp(-80z)
(5.51whose complete solution is
-Sz
2ipD(2p
-k2)
e-80z
(p,z)
_(-p,z)
E(p)
e8z
+
El(p)
e+
(5.6)
p2_
pg
386 P.S. DESHWAL and K.K.
MANN
(p z)
-
(-p z) 2EsinhBz +
4ip[(2p2_k2)6q
G(p)f(p)
+
D(2p 2-k2)(2p
k2)
+
2DB0k26]e
-Bz
-B0z
-Bz)
2_p)
+
2ipD(2p
k2)(e
e/(p
Differentiating (5.7) w.r.t, z and putting z 0, we find’(p)
’(-p)
2BE
4ip8f(p)[(2p2_ k2)6
q G(p)+
2DB0k2
(5.7)
2ipD(B-B
0)(2p
k2)
+
D(2p 2- k2)(2p
k2)]
+
(5.8)
p2_p
2E is obtained from here as the left hand member is known from
(4.8)
and(4.20).
We find the integrand in(5.3)
to be2ipD(B-B0)
(2P
-k2)
sinhBz
#_(p,z)
_(-p,z)
p2_p)
B
2ipD
(2p-k
2-80z
-Bz
+
(e e+
4ip[q(2p 2-k2)6G(p)
p2_p
-8z
+
D(2p2-k2)(2p-k2)
+
2Dk2B06]
e (5.9) The pole at pP0
contributes4P060
(2p-k
2)
G(p
0)
+
2DB0(2p+k2)]
e-ip0x
0
zl(X,Z)
[0)
[q
e (5.10)This represents the reflected wave in the region x < 0. This does not depend upon the width a of the mountain. The reflected shear wave in the region is found to be
4P0B
0-iP0x
-6 zl(X’Z)
[0)
[(2p-k
2)
F(Po)
4iPoBoD(2p-k2B060
)]
e e 0(5.11)
The integrand in(5.9)
has branch points at p k and p k. The branch cuts are given by the conditionsRe(B)
0Re(6).
As discussed byEwtng
and Press(1957),
the parts of branch cuts are hyperbolic as shown in figure 2. For contribution along the branch cut we put p+
iu, u being small as the main contribution is around the branch point. Along the cut,Re(B)
0 andIm(8)
changes signs along two sides of the cut. SinceB
is imaginary,82
is negative. Therefore82
=(
+ iu)2_
()2
2iU(l
+
i2)
U2(5.12)
From here
B
+
+i
i
0(5.13)
Integrating (5.3) along two sides of the branch cut, we find
k2x
i e
I’[
_(p,z) -_(-p,z)]B: i
#2
(x’z)
2 0
--_
e-UX
-[_(p
z)
(-p z)]
B
_i]
du2ek2
xf
[H (u)
sin zRAYLEIGH WAVE SCATTERING AT THE
FOOT OFA MOUNTAIN
387u is small,
Hi(u)
andH2(u
are expanded around u 0 and onlyHI(0)
andH2(0)
are retained. The following Laplace integrals(Oberhettinger,
(1973))
are usedsin
a
e-pt dtaCn
exp(-a2/4p)/2p
3/2
(5.15)
cos at/{--e
-pt dt(p/2
a2/4) exp(-a2/4p)/p
5/2
(5.16)
The scattered waves for the region x <0 are obtained to be2
(x,z)
(2k--2)
312
z
[(2(k2)2
+
k2)
+
2(2)2(2)2
+
k2).
x(x
+
oz
2)
5/2
x[q
G()(2(k--2)2
+
k2)
/((2
)2
+
k2)
280Dk2(2)2
+
k2)
iD(2p
k2)(z(2)2+
k2)].
exp[k--2
(x+z
2
/2x)
(5.17)
when x
>>z,
thenr
/(x
2+
z2)
x+z
2/2x
(5.18)
and the scattered wave is of the form
2
(x,z)
GO
exp(2r)/r
(5.19)
This represents a cylindrical wave. On the free surface
(z--0),
the wave has the behaviour ofexp(k2x)/x
3/2
We now find the waves transmitted to the other side of the mountain. The poten-tial for the region x > a is given by
_+ia -ipx
(x,z)
f_o+iao
+a(P’Z)
e dp(5.20)
o
The
countour is in the lower half of the complex plane in whichLa(-p,z)
is ana-lytic and so.+ie 2ipa
e-ipx
0
J_+ieo
+a
(-p’z)
e dpo From
(5.20)
and(5.21),
we obtain(5.21)
.+ie
[+a(p
z+a(-p,z)]e
dp(5.22)
(x,z)
-
=+ieo
e2ipa-
-ipxo
We take the Fourier transform of the wave equation
(2.3)
from x a to x tofind out
d2+a/dZ2
82+a
(a/ax)
a e ipa
ip()
a eipa
(5.23)
Changing p to -p and subtracting the resulting equation from(5.23)
it is found that(d2/dz
282)[+a
(P
z)e
-ipa+a(-p,z)e
ipa]
iPoa
-80z
2ipD(2p2-k
2)
e e(5.24)
388
8z
+a
(p’z)
e2ipa.+a(-p,z)
Dl(p)
e2ipD(2p-k
2 e ep2_
p02
Using
(4.22)
and(4.23)
and the procedure of(5.6),
we findi
(P+P0)
a-80z
2ipD(2pg-kZ)e
e2ipa
(-p,z)
2D sinhBz
+a(P’Z)
eT+a
p2_p
where
+
g(p)
exp(-Bz)/f(p)
-8z
+
D 2(p)e
i(p+p0)a
-80z
-(5.25)
(5.26)
g(p)
+2ip[-(2p2-k
2)
qG(p)
(l+e2ipa)
2ipBF(p)
(e2ipa-l)
2ipa
+
2pB0D(2p-k2)(p
0
+
_p0
2ipa
+
2P0
B0D(2p2-k
2)( I
eP+P0
P-P0
e2ip
a2p(2p-k2)BD(pO
+
e2ipa
2(2p-kP0B00D(p--O__
php
(5.27)
D in
(5.26)
is eliminated by the procedure for E in(5.7).
It is obtained thatLa(p,
z)
e2ipa--+a
(-p,z)
sinh8
Bz
[8F(p)
(l_e2ipa)
e2ipa
+iD(mp-k2)
(B-B0)
(P0
+
"P-P0
-)
i
(p+p0)a
-80z
-2ipm(2pg-k
2)
e e/
(pZ_pg)
+
g(p)exp(-Bz)/f (p)
(5.28)
The pole P
-P0
in the integrand in(5.22)
contributesiP0x
-80z
3(x,z)
-m(2pg-k
2)
e e4P0[q0(2pg-k2)
G(-P0)Cos2P0a
iP0
(x-a)
-B0z
-2P0B00qF(-P0)
sin2P0a
e e/f’
(-p0)
(5.29)
The first term cancels exactly the incident wave and the second term is the transitional wave in the region x > a.6.
CONCLUSIONS.
The integrals along the branch cuts_at p
-,
k lead to the scattered waves as obtained in(5.19).
They behave asexp(-2r)/r
where r is the distance from the scatterer. These are cylindrical waves which die at distant points from the foot of the mountain. On the free surface(z
0),
the scattered waves have the formexp(-2x)/x3/2
which is large at the points near the scatterer. Thus the energy of the scattered wavesRAYLEIGH WAVE SCATTERING AT THE
FOOT OFA MOUNTAIN
389transmitted wave in
(5.29)
depends upon width a of the mountain. As the distance from the other end of the mountain increases, the transmitted wave decreases exponentially anddies out at distant points. The reflected waves are given by (5.10) and (5.11) which do
not depend upon the width of the mountain. Numerical results for the amplitude of the scattered wave have been computed
or
Poisson’s solids for which k3
k at a point(r
I/2
km, z0)
in the region < 0 of the free surface. The results are obtained for q 0 and 1.8932 k. There is a sharp increase in the amplitude (fig.3)
when the wave number k increases through small values.The results of this paper have their application to underground nuclear explosions carried out on either side of the mountains llke Himalayas. It helps calculating the amount of energy reflected and scattered at the foot of the mountain and the amount of energy which is transmitted to the other side of the mountain.
X
z
C
6O
5O
4O
3O
2O
I0
/
-I0
-II
.12
-13
-14
-15
K
2
REFERENCES
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DESHWAL,
P.
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Geophs.,
85(1971),
107-124.2.
DESHWAL,
P.
S. Diffraction of Compressional Waves by a Strip in a LiquidHalf-space,
Jr.M@.
th.Phys._ Sci., 15(1981),
263-281.3.
EWING, W.
M.
JARDETSKY, W. S.,
andPRESS, F.
ElasticWaves
in Layered Media, McGraw Hill BookCo.,
1957.4.
KAZI, M.
H. The Lovewave Scattering Matrix for a Continental Margin, Geophys.J. R.
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Soc.,
52(1978),
25-44.
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MOMOI,
T. Scattering of the Rayleigh Wave by a Semi-circular RoughSurface, J.
Phys. Earth,28(1980),
497-519.6. MOMOI, To
Scattering of Rayleigh Waves by a Rectangular RoughSurface,
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295-319.
7.
NOBLE, B.
Methods Based on the Wiener-Hopf Technique, 1958.8.
OBERHETTINGER, F.
andBADII,
L. Tables ofLaplace transforms,
Springer-Verlag, NewYork, 1973.